Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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INVARIANT MEASURE AND THE EULER CHARACTERISTIC OF PROJECTIVELY ELAT MANIFOLDS

  • Jo, Kyeong-Hee (School of Mathematical Sciences Seoul National University) ;
  • Kim, Hyuk (School of Mathematical Sciences Seoul National University)
  • Published : 2003.01.01
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Abstract

In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP$^{n}$ invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that the Chen's conjecture is true for a closed affinely flat manifold whose holonomy group action permits an invariant probability Borel measure on RP$^{n}$ ; that is, such a closed affinly flat manifold has a vanishing Euler characteristic.

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References

  1. Linear Operators N. Dunford;J. T. Schwartz
  2. Comment. Math. Helv. v.55 Closed similarity manifolds D. Fried https://doi.org/10.1007/BF02566707
  3. Comm. Math. Helv. v.56 Affine manifolds with nilpotent holonomy D. Fried;W. Goldman;M. Hirsch https://doi.org/10.1007/BF02566225
  4. Contemp. Math. v.74 Geometric structures on manifolds and varieties of representations W. M. Goldman https://doi.org/10.1090/conm/074/957518
  5. Van Nostrand Math. Studies #16 Invariant means on topological groups and their applications F. P. Greenleaf
  6. Ann. of Math. v.10 Foliated bundles, invariant measures and flat manifolds M. W. Hirsch;W. P. Thurston
  7. Lecture notes in Math. v.1000 Differential Geometry in the Large H. Hopf
  8. Topology Appl. v.40 The Euler characteristic of a certain class of projectively flat manifolds H. Kim;H. Lee https://doi.org/10.1016/0166-8641(91)90051-M
  9. Proc. Amer. Math. Soc. v.118 The Euler characteristic of projectively flat manifolds with amenable fundamental groups H. Kim;H. Lee https://doi.org/10.2307/2160043
  10. Proceedings of the international conference on pure and applied Math., Beijing and Yanji A Polyhedral Gauss-Bonnet formula and projectively flat manifolds H. Kim
  11. Ist. Naz. Alta. Mat. Symp. Math. v.26 Invariant distances for projective structures S. Kobayashi
  12. Bull. Amer. Math. Soc. v.81 The Euler characteristic of a compact affine space form is zero B. Kostant;D. Sullivan https://doi.org/10.1090/S0002-9904-1975-13896-1
  13. Adv. Math. v.25 On fundamental groups of complete affinely flat manifolds J. Milnor https://doi.org/10.1016/0001-8708(77)90004-4
  14. Osaka J. Math. v.11 The affine structures on the real two-torus T. Nagano;K. Yagi
  15. The geometry and topology of 3-manifolds W. P. Thurston
  16. Monographs in Mathematics. v.81 Ergodic theory and semisimple groups R. J. Zimmer

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